Fast Hardware Design of 1/F Fractional Brownian Noise Generator G
Total Page：16
File Type：pdf, Size：1020Kb
JOURNAL OF COMMUNICATIONS AND INFORMATION SYSTEMS, VOL. 1, NO. 26, APRIL 2011 13 Fast Hardware Design of 1/f Fractional Brownian Noise Generator G. Loss and R. Coelho Abstract—This paper proposes the design of a fast and accurate signal processing and other scientific areas concerned with 1/f fractional Brownian motion (fBm) noise generator. The noise issues. Hardware architectures for white Gaussian noise solution is based on the successive random addition algorithm generators [11] [12] [13] have been reported in the literature. using the midpoint displacement (SRMD) technique. The imple- mentation was performed on a high-speed field-programmable However, from the authors knowledge, the proposition of a gate array (FPGA) Development Kit. It enabled the generation fast hardware design of 1/f fBm noise generator is addressed of 150 million of 16-bit noise samples per second. The 1/f noise for the first time in this work. This 1/f noise solution de- generator achieved 7.4% of the logic elements, 52% of the RAM scribes a hardware design for the successive random addition memory and 1.6% of the ROM memory. The accuracy of the method using the midpoint displacement (SRMD) technique 1/f noise samples was evaluated for different sequence size and confidence intervals. The noise samples pattern was examined introduced by Mandelbrot [9]. In a preliminary study [14] the by the probability density (PDF) and the heavy-tail distribution SRMD design was proposed and evaluated for the generation (HTD) functions. The results also include the auto-correlation of optical signal samples. The results showed to be very function (ACF) to show the low-frequency statistics of the 1/f promising. noise samples. For the investigation it was used the real 1/f The fractional noise generator was implemented on Altera speech-babble acoustic noise parameters. The results showed that the proposed 1/f noise generator design can be promising to the Stratix EP1S25 FPGA Development Kit using the Quartus performance evaluation of communications channels with low II v5.0 suite for Linux. The design proposed in this work bit error rate (BER) values. It can also be interesting for signal also considered the trade-off between the device resources processing applications and other areas concerned with noisy usage (e.g., memory, logic elements and embedded multipli- conditions. ers) to produce fast and accurate 1/f 16-bit noise samples. The design also includes a Gaussian random variables (RVs) Index Terms—1/f noise, fractional Brownian noise, low- frequency statistics, field-programmable gate array. block generator. The Box-Muller [15] method is the most important approach used for the generation of Gaussian RVs. The generation is based on transformation functions performed I. INTRODUCTION on Uniform variables. In [11] the authors proposed a hard- HE provision of fast and accurate noise generators has ware architecture based on the Box-Muller algorithm and the T become an important issue for digital communications Central Limit Theorem (CLT) [16]. This solution was also systems which attained reliable and high-speed channels. performed by Xilinx [17] and evaluated under different FPGA Moreover, 1/f noise [1] has been observed in several physical devices by the authors in [12] [13]. The results demonstrated systems such as, communication channels [2], electronic com- that the implementation based on both Box-Muller algorithm ponents and semiconductors devices [1] [3], natural phenom- and the CLT achieves accurate and fast Gaussian RVs. Hence, ena (e.g., rivers, ocean flows, average seasonal temperature, this method was used for the Gaussian samples generation. rainfall) [4], financial markets [5], image [6], acoustics and Nevertheless, the solution presented in this work introduces a music [7]. The 1/f noise is a random process characterized memory optimization for the Box-Muller functions evaluation. by its low-frequency or scaling invariance over a wide range The Tckacik algorithm [18] was applied for the Uniform of timescales [8]. White noise represented by the Gaussian RVs generation. This technique achieved fast generation, good distribution, is widely employed since it enables tractable randomness and few device resources requirements. analysis. But, since the presence of 1/f noise is a reality, The proposed design performance was examined in terms its investigation became a key issue. The fractional Brownian of its device resources requirements and the representation of motion (fBm) [8] [9] [10] leads to an efficient representation the pattern and low-frequency statistics (i.e., evaluations of the of the 1/f or fractional noise samples with low-frequency probability density (PDF), heavy-tail distribution (HTD) and statistics or scaling invariance degree represented by the Hurst auto-correlation coefficient (ACF) functions) of the 1/f noise (0 <H< 1) parameter [4]. samples. For the validation experiments, it was considered Therefore, the main contribution of this paper is the proposal the mean (m), standard-deviation (σ) and scaling parameter of a fast design of 1/f noise generator based on the fractional (H) estimated from the real 1/f factory2 acoustic noise of the Brownian motion. This enables more realistic and accurate NOISEX-92 [19] database. The implementation also included estimation of low bit error rate (BER) (10−9 to 10−12) a pure-Brownian noise, i.e., with m =0, σ =1 and H =1/2. in communications channels. It should be also applied to The results accuracy is examined and presented considering different samples sequence sizes and confidence intervals (CI) G. Loss and R. Coelho are with the Electrical Engineering Depart- ment, Military Institute of Engineering (IME), Rio de Janeiro, Brazil, e- [20]. For the low-frequency statistics (H parameter) estima- mail:{gloss,coelho}@ime.eb.br. tion it was used the wavelet-based method [21] considering JOURNAL OF COMMUNICATIONS AND INFORMATION SYSTEMS, VOL. 1, NO. 26, APRIL 2011 14 Daubechies filters with 12 coefficients. II. 1/F FRACTIONAL BROWNIAN MOTION NOISE Generally, noises can be represented by the power spectral density fluctuations per a frequency interval. Or, similarly, by its variations over a time scale of order 1/f β (0 β 2). In the literature, white, red/brow and pink noises are≤ also≤ referred as 1/f 0, 1/f 2 and 1/f noises, respectively. The 1/f noise is a random process with power spectral density (S(f)) defined as S(f) = 1 where 0 β 2. f β ≤ ≤ Mandelbrot showed that 1/f fractional noises have S(f) defined by S(f)= f 1−2H with H parameter values frequently close to 1. Thus, H parameter is the exponent of the power spectral density of the 1/f fractional noises. The H parameter expresses the low-frequency statistics, i.e., the scaling or time- (a) invariance degree, of a random process. It can also be defined by the decaying rate of the auto-correlation coefficient function ρ(k) ( 1 <ρ(k) < 1) as k . A random process can be classified− by its H (0 <H<→1) ∞ parameter as 1 • Anti-persistent process (0 <H < ): The ACF varies ∞ 2 rapidly and k=−∞ ρ(k)=0. 1 • Short-rangeP dependence process (SRD) (H = 2 ): The ACF ρ(k) exhibits an exponential decay to zero, such ∞ that k=−∞ ρ(k)= c, where c> 0 is a finite constant. 1 • Long-rangeP dependence process (LRD) ( 2 <H < 1): The ACF ρ(k) is a slowly-vanishing function. It has slow variations meaning a strong dependence degree between samples that are far apart or ∞ ρ(k) = . Thus, k=−∞ ∞ the S(f) is concentrated in theP low-frequency spectrum. Harold Edwin Hurst [4] found the presence of low-frequency or LRD statistics (H =0.72 0.006) in several observations (b) of different natural phenomena.± Brownian motion is a well Fig. 1. 1/f Noise Generator: (a) SRMD Algorithm [9], (b) 1/f fBm-block known example of a SRD process while 1/f fractional noises, Design. by definition, are LRD random processes [9]. The fractional Brownian motion [8] [9] is a random process 2 (XH (t)) with Gaussian independent increments, indexed by then, V ar[X(1) X(0)] = σ and V ar[X(t2) X(t1)] = the single scalar H parameter defined in with zero mean and t t σ2 for 0− t t 1. To achieve this− property a ℜ | 2 − 1| ≤ 1 ≤ 2 ≤ continuous sample path (null at origin). The variance of the random offset displacement (Di) with mean zero and variance increments is proportional to its time interval as V ar[X(t ) 2 2 −(i+1) 2 2 δi must be added to the samples where δi =1/2 σ . For 2H − X(t1)] α t2 t1 for 0 t1 t2. For all t1 and t2 instants example, the X(1/2) value is obtained by the interpolation | − | ≤ ≤ it follows that: XH (t) has stationary increments; XH (0) = 0 of X(0) and X(1) with variance δ2/22H+1. Then, several and E[XH (t)] = 0 for any instant t ; and XH (t) presents iterations are proceeded to compose the fBm random sequence. continuous sample paths. This means that fBm is a self-similar In order to provide stationary increments, after the midpoints n 2H random process, i.e., its statistical characteristics hold for any interpolation, a Di of a certain variance ( (r ) ; where r ∝ time scale. For any τ and r > 0, it follows that [XH (t + is the scaling factor.) is applied to all points (time increments) d −H and not just the midpoints. The X[i] noise sequence has τ) XH (t)τ≤0] ∼= r [XH (t + rτ) XH (t)τ≤0], where r is − d − i = N + 1 time increments, where N = 2maxlevel and the scaling factor and ∼= means similar in distribution.